Generalization of 𝐶*-algebra methods via real positivity for operator and Banach algebras

Abstract

Dedicated with affection and gratitude to Richard V. Kadison.Abstract. With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain C *algebraic results and theories to more general algebras. As motivation note that the 'completely' real positive maps on C * -algebras or operator systems are precisely the completely positive maps in the usual sense; however with real positivity one may develop a useful order theory for more general spaces and algebras. This is intimately connected to new relationships between an operator algebra and the C * -algebra it generates. We have continued this work together with Read, and also with Matthew Neal. Recently with Narutaka Ozawa we have investigated the parts of the theory that generalize further to Banach algebras. In the present paper we describe some of this work which is connected with generalizing various C * -algebraic techniques initiated by Richard V. Kadison. In particular Section 2 is in part a tribute to him in keeping with the occasion of this volume, and also discusses some of the origins of the theory of positivity in our sense in the setting of algebras, which the later parts of our paper developes further. The most recent work will be emphasized.1991 Mathematics Subject Classification. Primary 46B40, 46L05, 47L30; Secondary 46H10, 46L07, 46L30, 47L10.

Roots in Operator and Banach Algebras

“…That x 1 m is in the closed subalgebra generated by x may be found e.g. in the discussion after Proposition 6.3 in [6]. Now suppose that c 1 , c 2 are two mth roots of x with spectrum in S π m .…”

“…Note that the inequality (t1 + x) −1 ≤ 1/t for all t > 0 is equivalent to x being accretive (see e.g. [6,Lemma 2.4] It is well known that if the spectrum of an invertible element a contains no real strictly negative numbers then a is type M . Indeed for any a ∈ A the identity defining 'type M elements' is always true for t > 2 a by an inequality in the proof of the Neumann series lemma:…”

“…For example our results, particularly probably the geometric mean, should be applicable to our ongoing study of 'real positivity' in operator algebras (see e.g. [9,10,11,8,6] and references therein) initiated by the first author and Charles Read.…”

“…For example our results, particularly probably the geometric mean, should be applicable to our ongoing study of 'real positivity' in operator algebras (see e.g. [9,10,11,8,6] and references therein) initiated by the first author and Charles Read.Turning to background and notation, it is common when studying roots to make the assumption that the spectrum contains no strictly negative numbers. Note that a singular matrix with no strictly negative eigenvalues, may not have a square root (for example, E 12 in M 2 ), or may have a square root but not have a square root in {x} ′′ (for example, E 12 in M 3 , which has many square roots including E 13 + E 32 ), or may have infinitely many square roots in {x} ′′ (for example, 0 in an algebra with 1991 Mathematics Subject Classification.…”

“…Note too that a ∈ c A implies that a r ∈ c A for such r. This is because F A is closed under such powers (see e.g. [6,Proposition 6.3]). Also, in an operator algebra if W (a) ⊂ S θ for θ < π then W (a r ) ⊂ S rθ for 0 ≤ r ≤ 1 (see e.g.…”