Results on norm ultrapowers, group C*-algebras, spectral tests for representations, and nuclear algebras

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Spectral minima in ultraproducts — Lemma 2.3. For self-adjoint P_n with 0 <= P_n <= R in unital C*-algebras C_n, the class P of (P_n) in the ultraproduct satisfies min spec P = lim along omega of min spec P_n.
• Unitary lifts — Lemma 2.4. Every unitary of an ultraproduct of unital C*-algebras is the class of a sequence of unitaries.
• The full group C*-algebra — Proposition 3.1. For a discrete group G, C*(G) is a unital C*-algebra with unitaries u_g; every homomorphism of G into the unitary group of a unital C*-algebra C extends to exactly one unital *-homomorphism from C*(G) to C.
• Elementary matrices generate SL_3(Z) — Lemma 5.2. SL_3(Z) is generated by the six matrices I + E_ij, i different from j.
• A representation that is not periodic — Lemma 5.3. The group Z^3 semidirect SL_3(Z) has an irreducible finite-dimensional unitary representation pi_* that factors through reduction modulo 2, is nontrivial on translations, and is not equivalent to pi_* o theta^j for any j >= 1, where theta(v, M) = (2v, M).
• Spectral gap of the Laplacian — Lemma 1.2. Assume (T). For a homomorphism sigma of Gamma into the unitary group of a unital C*-algebra, the spectrum of sum over s in S of (sigma(s) - 1)*(sigma(s) - 1) lies in {0} union [kappa, 4|S|].
• The square-root trace inequality — Lemma 4.1. For positive Hilbert-Schmidt operators X, Y: |X - Y|_2^2 <= |X^2 - Y^2|_1 (Powers and Stormer).
• A uniform spectral bound — Lemma 5.1. Assume (T). If a unitary representation sigma of Gamma has no nonzero finite-dimensional invariant subspace, then for arbitrary unitaries U_s on any Hilbert space E, the Laplacian of the tuple (U_s tensor sigma(s)) is at least kappa^2/(4|S|).
• Nuclear finiteness — Proposition 8.1. Assume (T). For a tuple u of unitaries in a nuclear unital C*-algebra (no relations assumed), only finitely many classes of irreducible finite-dimensional unitary representations pi of Gamma have d(u, pi) < a.
• Induced representations by coset representatives — Lemma 3.1. For a subgroup H of G_0 with coset representatives q_i and a unitary representation alpha of H, the rule (rho(g)w) at j(g,i) = alpha(h(g,i)) w_i defines a unitary representation of G_0 whose restriction to H contains alpha in the coordinate of q_1 = e.
• Frobenius reciprocity, one direction — Lemma 3.2. For H of finite index, alpha irreducible and pi' an irreducible constituent of the induced representation, alpha occurs in the restriction of pi' to H.
• Predecessors — Proposition 5.1. Under a unital embedding of C*(G) into B^omega, the set of indices n for which every class in D_n has a predecessor in D_n is large.
• Finite chains of predecessors — Lemma 6.1. If D is a finite set of classes of irreducible finite-dimensional representations in which every pi has pi' with pi contained in pi' o theta, then every pi in D is equivalent to pi o theta^p for some p >= 1.
• No unital embedding into nuclear ultrapowers — Theorem 7.1. Assume (T) (property (T) of Z^3 semidirect SL_3(Z); not proved in the programme). For G = Z[1/2]^3 semidirect (SL_3(Z) x Z), every nuclear unital C*-algebra B and every free ultrafilter omega on N, there is no unital embedding of C*(G) into B^omega. With the nuclearity of O_2 (hypothesis (O)): a negative answer to Kirchberg's embedding problem.

Reading order

Three lessons, to be read in order. The theorem is proved from two named hypotheses that are not proved in the programme: property (T) of the group of integer translations and SL_3(Z), and (for the statement about O_2 only) the nuclearity of O_2. The facts on C*-algebras, trace-class operators and tensor norms are taken from two other courses of the programme and are listed at the places where they are used.