×

Similarity problems and length. (English) Zbl 0999.46025

This paper is a survey on the author’s recent results on R. V. Kadison’s conjecture in Am. J. Math. 77, 600-620 (1955; Zbl 0064.36605)]. The problem is the following:
Let \(A\) be any unital \(C^*\)-algebra, and \(u: A \mapsto B(H)\) (\(H\): Hilbert) any bounded unital homomorphism; does it exists an invertible element \(T \in B(H)\) such that: \(u_T: a \mapsto T^{-1}u(a)T\) is a \(\star \)-homomorphism,i.e. \(u\) is similar to a \(\star\)-homomorphism?
This problem has been solved in a certain number of cases but it stands still for the general situation.
U. Haagerup [Ann. Math., II. Ser. 118, 215-240 (1983; Zbl 0543.46033)], proved that \(u\) is similar to a \(\star\)-homorphism if and only if \(u\) is completely bounded (cb), which by definition means that the sequence \(( u \otimes I_n)_{n \in {\mathbb N}} \) is norm bounded where \(I_n\) is the identity map of \(M_n(\mathbb C)\), and \(\|u\|_{cb}\) \((= \{ \sup \|u \otimes I_n \|\)) is also equal to \(\inf \{\|T^{-1} \|.\|T \|, u_T\) is a \(\star\)-homomorphism}.
An extension of this last theorem has been given by V. I. Paulsen in J. Funct. Anal. 55, 1-17 (1984; Zbl 0557.46035), when \(A\) is any unital operator algebra (i.e. any closed unital subalgebra of some \(B(\mathcal H)\)). So G. Pisier gives a generalization, namely (SP), of Kadison’s problem:
For any operator algebra \(A\) and any bounded homomorphism \(u: A \mapsto B(H)\) (\(H\) an arbitrary Hilbert space), is \(u\) cb? This last conjecture is false as the author gives a counter-example using the disc algebra \(A(\mathbb D)\), which is the completion of the set of numerical polynomials \(P\) for the norm \(\|P \|_\infty = \sup \{|P(z)|\;|z \in \partial(\mathbb D) \}\).
G. Pisier gives a tight relation between (SP) and a notion of length for operator algebras. By theorem 5, a unital operator algebra satisfies (SP) if and only if there exist \(d \in \mathbb N,K \in \mathbb R^+\), such that \(\forall n \in \mathbb N\), \(\forall x \in M_n(A)\) (the \(n \times n\) matrices with entries in \(A\)), there exist a natural number \(N(n,x)\), and scalar matrices: \(\alpha_0 \in M_{nN}(\mathbb C)\) (rectangular matrices), \( \alpha_1,\dots,\alpha_{d-1} \in M_n(\mathbb C),\;\alpha_d \in M_{Nn}(\mathbb C)\), together with diagonal matrices \(D_1,\dots,D_d\) in \(M_N(A)\) satisfying: \[ x = \alpha_0D_1\alpha_1D_1\dots D_d\alpha_d,\tag{1} \]
\[ \prod_0^d \|\alpha_i \|\prod_1^d \|D_i \|\leq K \|x \|.\tag{2} \] Moreover, If one denotes by \(l(A)\) the smallest integer \(d\) for which this holds, one has: \(l(A) = \inf \{\alpha \mid \exists C, \forall u\) unital homomorphism \(A \mapsto B(H),\) \(\|u\|_{cb} \leq C\|u\|^{\alpha }\},\) and this last extremum is actually a minimum. Then giving explicit values for their length, the author recovers the known examples for which (SP) is true.
Using the length condition, Proposition 9 says that initial Kadison’s conjecture is always true if and only if there exists a universal constant \(d_0\) such that any unital \(C^*\)-algebra has a length less or equal to \(d_0\). To end the survey, the author proves that this last universal condition is not true for its generalization to (non-self-adjoint) operator algebras. In fact, for any natural number \(d\) there exists an operator algebra \(A_d\) such that \(l(A_d) = d\).

MSC:

46L07 Operator spaces and completely bounded maps
47L55 Representations of (nonselfadjoint) operator algebras
47A65 Structure theory of linear operators
47D99 Groups and semigroups of linear operators, their generalizations and applications
PDFBibTeX XMLCite
Full Text: DOI arXiv