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Weighted shifts and the construction

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson builds the algebra \(B\) of Theorem 3.2 of Quasitraces and a uniform additivity defect from the bundle tests of Bundle tests with a Chern class obstruction. Each test, at each point of its base, gives \(24\) weighted shifts on the Hilbert space \(E_0\oplus E_1\oplus\cdots\) of its fibers, and \(B\) is generated by the families of these shifts over all tests and all points. The two relations hold level by level. A pair of isometries with orthogonal ranges in \(M_m(B)\) can be approximated by a matrix of polynomials of degree at most \(R\); in the test with parameters \((m,R)\), restricted to level zero, such a matrix produces a trivial subbundle of rank \(2md_0\), which the test excludes.

The lesson follows [OpenAI-QT, Section 5]. From the lesson Order, local units and quotients of C*-algebras (course Foundations of von Neumann algebras) it uses the description of a generated C*-subalgebra by polynomials (Definition 3.1 there), spectral permanence (Theorem 3.2 there) and the positive cone (Theorem 8.2 there): a self-adjoint element is positive exactly when it has the form \(y^*y\).

1. Bounded products

Let \((H_\omega)_{\omega\in\Omega}\) be a family of Hilbert spaces, and let \(D=\prod_\omega\mathcal B(H_\omega)\) be the set of families \(a=(a_\omega)\) of bounded operators with \(\|a\|=\sup_\omega\|a_\omega\|<\infty\), with the coordinatewise operations and adjoint. Acting coordinatewise on the Hilbert direct sum \(\bigoplus_\omega H_\omega\), the families are exactly the bounded operators that map each \(H_\omega\) into itself, with the same norm. They form a norm-closed subalgebra of \(\mathcal B\bigl(\bigoplus_\omega H_\omega\bigr)\) that is closed under the adjoint, so \(D\) is a unital C*-algebra.

Positivity. A self-adjoint \(a\in D\) is positive if and only if every \(a_\omega\) is positive. Indeed, if \(a_\omega=y_\omega^*y_\omega\) with \(y_\omega=a_\omega^{1/2}\), then \(\|y_\omega\|=\|a_\omega\|^{1/2}\le\|a\|^{1/2}\), so \(y=(y_\omega)\in D\) and \(a=y^*y\). Conversely, if \(a=y^*y\), then \(a_\omega=y_\omega^*y_\omega\). The same argument applies to an operator that is block diagonal for an orthogonal decomposition of one Hilbert space: it is positive if and only if its blocks are. If \(B\subseteq D\) is a C*-subalgebra containing the unit, a self-adjoint element of \(B\) has the same spectrum in \(B\) and in \(D\) by spectral permanence, so it is positive in \(B\) if and only if it is positive in \(D\).

Matrices. For \(n\ge1\), an \(n\times n\) matrix over \(D\) is a family of operators on the spaces \(H_\omega^n\), bounded because its norm is at most the sum of the norms of its entries. So \(M_n(D)=\prod_\omega\mathcal B(H_\omega^n)\), and for a unital C*-subalgebra \(B\subseteq D\), \(M_n(B)\) is the C*-subalgebra of the matrices with entries in \(B\). Rectangular matrices \(P\in M_{m,n}(B)\) act as families of operators \(H_\omega^n\to H_\omega^m\); then \(P^*P\in M_n(B)\), and \(\|P\|\) is at most the sum of the norms of the entries. For an element \(a\) of \(D\), or a matrix over \(D\), we write \(a(\omega)\) for its coordinate at \(\omega\).

2. Weighted shifts

Fix integers \(m,R\ge1\) and data as in Theorem 6.1 of Bundle tests with a Chern class obstruction: a compact smooth manifold \(Y\), Hermitian bundles \(E_j\) over \(Y\) for \(j\ge0\), with \(E_0\) trivial of rank \(d_0=64m4^R\), and bundle maps \(X_i^{(j)}\colon E_j\to E_{j+1}\) for \(1\le i\le24\). For \(y\in Y\) let \[ H_y=\bigoplus_{j=0}^\infty E_j(y) \] be the Hilbert direct sum of the fibers: the sequences \(\xi=(\xi_j)\) with \(\xi_j\in E_j(y)\) and \(\sum_j\|\xi_j\|^2<\infty\). The fiber \(E_j(y)\) is the level \(j\) of \(H_y\). For \(1\le i\le24\) let \(S_i(y)\) be the weighted shift \[ \begin{gathered} \bigl(S_i(y)\xi\bigr)_0=0,\\ \bigl(S_i(y)\xi\bigr)_{j+1}=X_i^{(j)}(y)\,\xi_j\quad(j\ge0) . \end{gathered} \]

Lemma 2.1. \(S_i(y)\) is a bounded operator of norm at most \(1\), its adjoint is \[ \bigl(S_i(y)^*\xi\bigr)_j=X_i^{(j)}(y)^*\,\xi_{j+1}\qquad(j\ge0), \] and \[ \begin{gathered} \sum_{i=1}^{24}S_i(y)^*S_i(y)=I,\\ \sum_{i=1}^{24}S_i(y)S_i(y)^*\le\tfrac23\,I . \end{gathered} \]

Proof. We omit \(y\). The column relation \(\sum_iX_i^{(j)*}X_i^{(j)}=I\) gives \(X_i^{(j)*}X_i^{(j)}\le I\), so \(\|X_i^{(j)}\|\le1\) and \(\|S_i\xi\|^2=\sum_j\|X_i^{(j)}\xi_j\|^2\le\|\xi\|^2\). The formula for the adjoint follows from \[ \begin{aligned} \langle S_i\xi,\zeta\rangle&=\sum_{j\ge0}\langle X_i^{(j)}\xi_j,\zeta_{j+1}\rangle\\ &=\sum_{j\ge0}\langle\xi_j,X_i^{(j)*}\zeta_{j+1}\rangle . \end{aligned} \] So \(S_i^*S_i\) acts on level \(j\) by \(X_i^{(j)*}X_i^{(j)}\), and \(S_iS_i^*\) acts by \(0\) on level \(0\) and by \(X_i^{(j)}X_i^{(j)*}\) on level \(j+1\). Both are block diagonal for the decomposition into levels. Summing over \(i\), the first relation of Theorem 6.1 gives \(\sum_iS_i^*S_i=I\). The operator \(\tfrac23I-\sum_iS_iS_i^*\) is block diagonal with the blocks \(\tfrac23I\) on level \(0\) and \(\tfrac23I-\sum_iX_i^{(j)}X_i^{(j)*}\) on level \(j+1\), which are positive by the second relation; by Section 1 it is positive. \(\blacksquare\)

3. The algebra

For every pair of integers \(m,R\ge1\) fix data as in Section 2, with base \(Y_{m,R}\). Let \(\Omega\) be the set of the triples \(\omega=(m,R,y)\) with \(m,R\ge1\) and \(y\in Y_{m,R}\); write \(H_\omega=H_y\) and \(S_i(\omega)=S_i(y)\) for the data of the test \((m,R)\). Let \(D=\prod_{\omega\in\Omega}\mathcal B(H_\omega)\), and \[ x_i=\bigl(S_i(\omega)\bigr)_{\omega\in\Omega}\in D\qquad(1\le i\le24), \] which is an element of \(D\) because \(\|S_i(\omega)\|\le1\) for all \(\omega\). Let \(B=C^*(1,x_1,\dots,x_{24})\subseteq D\).

Lemma 3.1. \(B\) is a separable unital C*-algebra, and \[ \sum_{i=1}^{24}x_i^*x_i=1,\qquad\sum_{i=1}^{24}x_ix_i^*\le\tfrac23\cdot1 . \] The noncommutative polynomials in \(x_1,\dots,x_{24},x_1^*,\dots,x_{24}^*\) are dense in \(B\).

Proof. The relations hold in every coordinate by Lemma 2.1, so the first holds in \(D\) and the second in \(D\) by the positivity criterion of Section 1. Both sides lie in \(B\), so the second holds in \(B\) as well. By Definition 3.1 of Order, local units and quotients of C*-algebras, \(B\) is the closure of the noncommutative polynomials in the \(x_i\) and \(x_i^*\), with complex coefficients and a constant term. The polynomials with coefficients in \(\mathbb Q+i\mathbb Q\) form a countable set, which is dense. \(\blacksquare\)

The index set \(\Omega\) is uncountable, and \(D\) is far from separable; \(B\) is separable because it has finitely many generators. Different tests and different points of a base are not related by any continuity in \(D\). Continuity in the point \(y\) is used only inside a single test, through the bundle maps.

4. Finite propagation

A word of length \(l\ge0\) is a product \(z_1z_2\cdots z_l\) of letters \(z_k\in\{x_1,\dots,x_{24},x_1^*,\dots,x_{24}^*\}\); the empty word is \(1\). A polynomial has degree at most \(R\) if it is a linear combination of words of length at most \(R\).

Lemma 4.1. Let \(\omega=(m,R,y)\in\Omega\) and let \(w\) be a word of length \(l\le R\). Then \(w(\omega)\) maps level \(0\) of \(H_\omega\) into a single level \(k\le l\), and its restriction to level \(0\) is the value at \(y\) of a composition of bundle maps \(X_i^{(j)}\) and \(X_i^{(j)*}\) with \(j<l\), or is zero.

Proof. Induction on \(l\). The empty word is the identity of level \(0\). Let \(w=zw'\), where \(w'\) has length \(l-1\) and maps level \(0\) into level \(k\le l-1\) by such a composition. If \(z=x_i\), then \(z\) maps level \(k\) to level \(k+1\le l\) by \(X_i^{(k)}\), with \(k<l\). If \(z=x_i^*\), then \(z\) maps level \(k\ge1\) to level \(k-1\) by \(X_i^{(k-1)*}\), and level \(0\) to \(0\). \(\blacksquare\)

So a word of length at most \(R\) involves only the levels \(0,\dots,R\) and the maps \(X_i^{(j)}\) with \(j<R\); the continuation of the test beyond level \(R\) plays no role. Consequently, let \(P\in M_{m,2m}(B)\) be a matrix whose entries are polynomials of degree at most \(R\), and \(\omega=(m,R,y)\). Then \(P(\omega)\) maps \(E_0(y)^{2m}\), the level \(0\) part of \(H_\omega^{2m}\), into \[ F(y)=\bigl(E_0(y)\oplus\dots\oplus E_R(y)\bigr)^m\subseteq H_\omega^m , \] and its restriction is the value at \(y\) of a bundle map \[ P_0\colon E_0^{\oplus2m}\to F=\bigl(E_0\oplus\dots\oplus E_R\bigr)^{\oplus m} \] over \(Y_{m,R}\). It is continuous, because each of its blocks \(E_0\to E_k\) is a finite linear combination of compositions of the continuous bundle maps \(X_i^{(j)}\) and \(X_i^{(j)*}\). With the inclusions \(\iota_0\colon E_0(y)^{2m}\to H_\omega^{2m}\) and \(\iota_F\colon F(y)\to H_\omega^m\), this says \[ P(\omega)\,\iota_0=\iota_F\,P_0(y) . \]

5. No properly infinite matrix algebra

Theorem 5.1. For every \(m\ge1\), \(M_m(B)\) is not properly infinite.

Proof. Suppose that \(s_1,s_2\in M_m(B)\) satisfy \(s_k^*s_l=\delta_{kl}\,1\). The \(m\times2m\) matrix \(W=(s_1\ s_2)\in M_{m,2m}(B)\) satisfies \(W^*W=I_{2m}\), so \(\|W\|=1\). By Lemma 3.1, every entry of \(W\) can be approximated by polynomials in the \(x_i\) and \(x_i^*\); approximating each of the \(2m^2\) entries within \(1/(10m^2)\) gives \(P\in M_{m,2m}(B)\) with polynomial entries and \(\delta=\|P-W\|\le\tfrac15\). Then \[ \begin{aligned} P^*P-I_{2m}&=(P-W)^*P\\ &\quad+W^*(P-W),\\ \|P^*P-I_{2m}\|&\le\delta(1+\delta)+\delta\\ &=2\delta+\delta^2\le\tfrac{11}{25}<\tfrac12 . \end{aligned} \] Let \(R\ge1\) be at least the length of every word that occurs in the entries of \(P\), and consider the test \((m,R)\). For \(y\in Y_{m,R}\) and \(\omega=(m,R,y)\), the coordinate \(P(\omega)^*P(\omega)-I\) has norm less than \(\tfrac12\), so \[ P(\omega)^*P(\omega)\ge\tfrac12\,I \] on \(H_\omega^{2m}\). By Section 4 and because \(\iota_F\) is isometric, \[ \begin{aligned} P_0(y)^*P_0(y)&=\bigl(\iota_FP_0(y)\bigr)^*\,\iota_FP_0(y)\\ &=\bigl(P(\omega)\iota_0\bigr)^*\,P(\omega)\iota_0\\ &=\iota_0^*\,P(\omega)^*P(\omega)\,\iota_0\ge\tfrac12\,I . \end{aligned} \] By Lemma 1.1 of Expanding maps and a continuous normalization, \(Q=P_0(P_0^*P_0)^{-1/2}\) is a bundle map with \(Q^*Q=I\), and it is an isomorphism from \(E_0^{\oplus2m}\) onto its image, a subbundle of \(F\). Since \(E_0\) is trivial of rank \(d_0\), this image is a trivial subbundle of rank \(2md_0\) of \(\bigl(E_0\oplus\dots\oplus E_R\bigr)^{\oplus m}\). This contradicts part 3 of Theorem 6.1 of Bundle tests with a Chern class obstruction. \(\blacksquare\)

Proof of Theorem 3.2 of Quasitraces and a uniform additivity defect. Lemma 3.1 and Theorem 5.1. \(\blacksquare\)

With the construction theorem, Section 5 of the same lesson proves the main theorem of the course, Theorem 3.1 there, under the hypotheses (CBH), for the existence of a quasitrace on \(A=M_{48}(B)\), and (SM), for the incidence lemma. The algebra \(B\) has no tracial state, and every normalized \(1\)-quasitrace on \(A\) has the additivity defect at least \(\tfrac1{144}\) on the pair \(a,b\); these two statements use only (SM).

The test with parameters \((m,R)\) excludes only pairs of isometries in \(M_m(B)\) that can be approximated, within \(\tfrac15\), by polynomials of degree at most \(R\). Every pair can be approximated in this way for some \(R\), and the algebra contains the tests for all \((m,R)\) at once. This is why the number of generators and the norm bounds have to be uniform over all tests, while the bases and the ranks grow with \(m\) and \(R\).

6. A sharper constant

Corollary 6.1. For every \(c\) with \(\tfrac12<c<1\) there is a separable unital C*-algebra \(B_c\), generated by \(1\) and seven elements \(x_1,\dots,x_7\), such that \[ \sum_{i=1}^7x_i^*x_i=1,\qquad\sum_{i=1}^7x_ix_i^*\le c\cdot1 , \] and \(M_m(B_c)\) is not properly infinite for any \(m\ge1\). For the positive contractions \(a,b\in M_{14}(B_c)\) of Lemma 4.1 of Quasitraces and a uniform additivity defect, every normalized \(1\)-quasitrace \(\tau\) on \(M_{14}(B_c)\) satisfies \[ \tau(a+b)-\tau(a)-\tau(b)\ge\frac{1-c}{14} , \] and, assuming (CBH), \(M_{14}(B_c)\) has normalized \(2\)-quasitraces.

Proof. By Remark 6.2 of Bundle tests with a Chern class obstruction, at each level \(j<R\) of each test there are seven bundle maps that double the dimension of every subspace. Remark 6.2 of Expanding maps and a continuous normalization, with \(\theta=1/c\), which lies strictly between \(1\) and \(2\), normalizes them to maps \(X_1,\dots,X_7\) with \(\sum_iX_i^*X_i=I\) and \(\sum_iX_iX_i^*\le c\,I\). Beyond level \(R\), the two halved inclusions have the row sum \(\tfrac12I\le c\,I\). The obstruction, part 3 of Theorem 6.1 there, does not depend on the maps. With these tests, Sections 2 to 5 apply verbatim with seven generators and the bound \(c\). Lemma 4.1 of Quasitraces and a uniform additivity defect, with \(N=7\), gives the defect; (CBH) gives the quasitraces as in Section 5 of that lesson. \(\blacksquare\)

So every constant below \(\tfrac1{28}\) occurs as a uniform additivity defect, compared with \(\tfrac1{144}\) for twenty-four generators and \(c=\tfrac23\).

7. Exercises

7.1. Show that a word \(w\) of length \(l\) maps level \(0\) into a level \(k\le l\) with \(k\equiv l\pmod 2\), or to zero.

7.2. Let \(C\) be a unital C*-algebra and \(s_1,s_2\in M_m(C)\). Show that the matrix \(W=(s_1\ s_2)\in M_{m,2m}(C)\) satisfies \(W^*W=I_{2m}\) if and only if \(s_1\) and \(s_2\) are isometries with \(s_1^*s_2=0\).

7.3. Let \(C\) be a unital C*-algebra and \(P\in M_{m,2m}(C)\) with \(\|P^*P-I_{2m}\|<1\). Show that \(M_m(C)\) is properly infinite.

7.4. Show that \(\sum_ix_ix_i^*\) is not invertible in \(B\), although \(\sum_ix_i^*x_i=1\).

7.5. Compute the constant \((1-c)/(2N)\) of the defect lemma for \(N=24\), \(c=\tfrac23\), and for \(N=7\), \(c=\tfrac23\), and its supremum for \(N=7\) and \(\tfrac12<c<1\).

8. Solutions

7.1. Each letter changes the level by \(+1\) or \(-1\), or maps to zero. After \(l\) letters starting at level \(0\), the level is \(l-2u\), where \(u\) is the number of steps down.

7.2. The \(2\times2\) block matrix \(W^*W\) has the entries \(s_k^*s_l\), \(k,l\in\{1,2\}\). It is \(I_{2m}\) exactly when \(s_k^*s_l=\delta_{kl}\,1\), that is, when \(s_1,s_2\) are isometries and \(s_1^*s_2=0\) (then \(s_2^*s_1=(s_1^*s_2)^*=0\)).

7.3. Put \(\delta=\|P^*P-I_{2m}\|<1\). The self-adjoint element \(P^*P\) has spectrum in \([1-\delta,1+\delta]\), so it is positive and invertible, and \(Q=P(P^*P)^{-1/2}\) satisfies \(Q^*Q=(P^*P)^{-1/2}P^*P(P^*P)^{-1/2}=I_{2m}\). By Exercise 7.2 the two \(m\times m\) blocks of \(Q\) are isometries with orthogonal ranges. So proper infiniteness is witnessed by approximate solutions, which is what makes the polynomial approximation in Theorem 5.1 possible.

7.4. At every coordinate \(\omega\), \(\sum_iS_i(\omega)S_i(\omega)^*\) vanishes on level \(0\), which is not zero since \(E_0\) has positive rank. An invertible element of \(D\) has invertible coordinates, so \(\sum_ix_ix_i^*\) is not invertible in \(D\), and not in \(B\).

7.5. \((1/3)/48=1/144\); \((1/3)/14=1/42\); \((1-c)/14\) increases to \(\tfrac12\cdot\tfrac1{14}=\tfrac1{28}\) as \(c\) decreases to \(\tfrac12\), and the supremum \(\tfrac1{28}\) is not attained.

References